# Law of Large Number and CLT

### The Weak Law of Large Numbers (WLLN)

If X

_{1}, X_{2}, . . . , X_{n}are IID, then

This theorem says that the distribution of X̄ becomes concentrated around µ as n gets large. For example, the proportion of heads in a large number of tosses is expected to be close to 1/2. This, however, doesn’t mean that X̄ will be numerically equal to 1/2. It means that, when n is large, the distribution of X̄ is tightly concentrated around 1/2.

**PROOF**:

Using Chebyshev’s inequality,

which tends to 0 as n → ∞. Note that the Variance of the Sample mean is σ^{2}/n. Find the proof here.

**Example:**

In tossing a coin, let p=0.5. How large should be n so that P(0.4 ≤ X̄ ≤ 0.6) ≥ 0.7?**Solution: **E(X̄_{n}) = p = 1/2 and V(X̄_{n}) = σ^{2}/n = p(1 – p)/n = 1/(4n)

P(0.4 ≤ X̄_{n} ≤ 0.6) = P(|X̄_{n} – µ| ≤ 0.1)

= 1 – P(|X̄_{n} – µ| ≥ 0.1)

≥ 1 – (1 / 4n(0.1)^{2}) = 1 – 25/n

which will be larger than 0.7 if n=84.

### Central Limit Theorem

The central limit theorem says that the sample mean X̄_{n} is approximately Normal with mean µ and variance σ^{2}/n.

If X

_{1}, X_{2}, . . . , X_{n}are IID with mean µ and variance σ^{2}. Let X̄_{n}be the sample mean then

where Z ~ N(0, 1).